# A Partial Integral Equation (PIE) Representation of Coupled Linear PDEs and Scalable Stability Analysis using LMIs

## TL;DR

This paper introduces a novel PIE representation for coupled linear PDEs enabling stability analysis via convex optimization, avoiding discretization, and demonstrating scalability up to 40 PDEs with practical implementation in MATLAB.

## Contribution

It provides a standardized PIE representation for coupled linear PDEs and develops convex optimization-based stability analysis methods that are scalable and implementable in PIETOOLS.

## Key findings

- PIE representation is equivalent to well-posed coupled PDEs.
- Convex optimization verifies stability without discretization.
- Algorithm scales to systems with up to 40 PDEs.

## Abstract

We present a new Partial Integral Equation (PIE) representation of Partial Differential Equations (PDEs) in which it is possible to use convex optimization to perform stability analysis with little or no conservatism. The first result gives a standardized representation for coupled linear PDEs in a single spatial variable and shows that any such PDE, suitably well-posed, admits an equivalent PIE representation, defined by the given conversion formulae. This leads to a new prima facie representation of the dynamics without the implicit constraints on system state imposed by boundary conditions. The second result is to show that for systems in this PIE representation, convex optimization may be used to verify stability without discretization. The resulting algorithms are implemented in the Matlab toolbox PIETOOLS, tested on several illustrative examples, compared with previous results, and the code has been posted on Code Ocean. Scalability testing indicates the algorithm can analyze systems of up to 40 coupled PDEs on a desktop computer.